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Term Structures of Implied Volatilities

This paper studies modelling and existence issues for market models of stochastic implied volatility in a continuous-time framework with one stock, one bank account and a family of
European options for all maturities with a fixed payoff function h. We first characterize absence of arbitrage in terms of drift conditions for the forward implied volatilities corresponding
to a general convex h. For the resulting infinite system of SDEs for the stock and all the forward implied volatilities, we then study the question of solvability and provide sufficient
conditions for existence and uniqueness of a solution. We do this for two examples of h, namely calls with a fixed strike and a fixed power of the terminal stock price, and we give explicit
examples of volatility coefficients satisfying the required assumptionsShow more